Hybrid Quantum-Classical Branch-and-Price for Intra-Day Electric Vehicle Charging Scheduling via Partition Coloring

Liang Zhong; Li Wang; Peng Sun; Qing-Guo Zeng

arxiv: 2603.21374 · v2 · submitted 2026-03-22 · 💻 cs.CE · math.OC

Hybrid Quantum-Classical Branch-and-Price for Intra-Day Electric Vehicle Charging Scheduling via Partition Coloring

Peng Sun , Liang Zhong , Qing-Guo Zeng , Li Wang This is my paper

Pith reviewed 2026-05-15 01:01 UTC · model grok-4.3

classification 💻 cs.CE math.OC

keywords electric vehicle chargingpartition coloringbranch-and-pricequantum annealingQUBOschedulingconflict graphhybrid optimization

0 comments

The pith

Hybrid quantum-classical branch-and-price solves large EV charging schedules to optimality where pure classical solvers time out.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models intra-day electric vehicle charging scheduling as a partition coloring problem whose vertices are candidate charging intervals for each vehicle and whose edges encode temporal and charger-capacity conflicts. It builds a branch-and-price algorithm whose restricted master problem is solved with Gurobi and whose pricing subproblem is reformulated as a QUBO and solved with quantum-annealing-inspired algorithms (ballistic simulated branching and simulated coherent Ising machines). On families of synthetic instances the hybrid versions match the pure Gurobi baseline on small and medium problems but close optimality gaps and prove optimality on large, hard instances within the same time limit. The result matters because fleet operators need fast, provably feasible schedules under tight charger limits, and the hybrid scheme suggests a practical route to scaling such combinatorial scheduling problems.

Core claim

Embedding QAIA solvers for the maximum-independent-set pricing subproblem inside classical branch-and-price produces an algorithm that matches Gurobi on small and medium EV instances and strictly outperforms it on large instances by closing gaps that the pure classical solver leaves open at the time limit.

What carries the argument

Branch-and-price decomposition on the partition-coloring conflict graph, with the pricing subproblem cast as a QUBO and solved by ballistic simulated branching or simulated coherent Ising machine while the master problem remains with Gurobi.

Load-bearing premise

The synthetic EV charging instances and the QUBO reformulation of their maximum-independent-set subproblems preserve the difficulty and solution quality of real-world intra-day scheduling.

What would settle it

Extract conflict graphs from actual operational logs of a public EV charging facility, run both the hybrid QAIA variants and the pure Gurobi branch-and-price on them under identical time limits, and observe whether the QAIA versions still close all gaps while the baseline does not.

Figures

Figures reproduced from arXiv: 2603.21374 by Liang Zhong, Li Wang, Peng Sun, Qing-Guo Zeng.

**Figure 1.** Figure 1: Schematic illustration of the EV charging system. The station contains two chargers, and four EVs are shown, [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: Illustration of candidate charging intervals. The horizontal axis represents time, and the vertical axis lists [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Construction of the conflict graph for candidate charging intervals. Subfigure (a) illustrates intra-vehicle [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: Overview of the proposed solution framework for the EV charging scheduling problem. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Scalability comparison of branch-and-price variants. Subfigure (a) MIP gap versus instance size, where [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

read the original abstract

The rapid deployment of electric vehicles (EVs) in public parking facilities and fleet operations raises challenging intra-day charging scheduling problems under tight charger capacity and limited dwell times. We model this problem as a variant of the Partition Coloring Problem (PCP), where each vehicle defines a partition, its candidate charging intervals are vertices, and temporal and resource conflicts are represented as edges in a conflict graph. On this basis, we design a branch-and-price algorithm in which the restricted master problem selects feasible combinations of intervals, and the pricing subproblem is a maximum independent set problem. The latter is reformulated as a quadratic unconstrained binary optimization (QUBO) model and solved by quantum-annealing-inspired algorithms (QAIA) implemented in the MindQuantum framework, specifically the ballistic simulated branching (BSB) and simulated coherent Ising machine (SimCIM) methods, while the master problem is solved by Gurobi. Computational experiments on a family of synthetic EV charging instances show that the QAIA-enhanced algorithms match the pure Gurobi-based branch-and-price baseline on small and medium instances, and clearly outperform it on large and hard instances. In several cases where the baseline reaches the time limit with non-zero optimality gaps, the QAIA-based variants close the gap and prove optimality within the same time budget. These results indicate that integrating QAIA into classical decomposition schemes are a promising direction for large-scale EV charging scheduling and related PCP applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gets better results on large synthetic EV instances by using QAIA heuristics for the pricing subproblem, but the optimality claims are shaky because those heuristics lack guarantees.

read the letter

The main thing to know is that this work takes a standard branch-and-price setup for EV charging, models it as partition coloring, turns the maximum independent set pricing problem into QUBO, and solves that subproblem with BSB and SimCIM from MindQuantum instead of an exact solver. On the synthetic test set the hybrid versions match Gurobi on small and medium cases and beat it on the large hard ones, sometimes closing gaps and reporting optimality where the baseline times out with positive gaps. That performance difference on bigger instances is the concrete result worth noting. The modeling is straightforward and the choice to plug quantum-inspired methods into the column-generation loop is a direct engineering move rather than a deep theoretical shift. The experiments are the part that stands out, and they suggest the hybrid approach can be useful for scaling decomposition methods on scheduling problems with dense conflict graphs. The soft spot is exactly what the stress-test flags: BSB and SimCIM are iterative heuristics without optimality certificates. Branch-and-price needs the pricing oracle to return a column with negative reduced cost whenever one exists, or certify that none does. If the QAIA returns a suboptimal independent set on a hard instance, the algorithm can terminate with an unproven gap even though better columns were available. The abstract claims the QAIA variants prove optimality in cases where Gurobi does not, but that only holds if the heuristics happened to hit the exact optimum every relevant time. The paper does not appear to include verification steps or a fallback to exact solving, so those optimality statements rest on an untested assumption. Everything is also synthetic, with no details yet on how the instances were generated or how representative they are of real charger networks. This is for people working on hybrid solvers for transportation and resource scheduling. A reader who cares about practical EV operations or column-generation tweaks could extract value from the computational comparison. I would send it to peer review so the subproblem accuracy and instance details can be checked properly.

Referee Report

2 major / 2 minor

Summary. The paper models intra-day EV charging scheduling as a Partition Coloring Problem on a conflict graph and develops a branch-and-price algorithm whose restricted master problem is solved by Gurobi while the pricing subproblem (maximum independent set) is converted to QUBO and solved by the heuristic QAIA methods BSB and SimCIM implemented in MindQuantum. On synthetic instances the QAIA-enhanced variants match the pure Gurobi branch-and-price baseline on small and medium cases and outperform it on large instances by closing optimality gaps and proving optimality within the same time limit.

Significance. If the reported optimality proofs are valid, the work demonstrates that quantum-annealing-inspired heuristics can be embedded inside classical exact decomposition schemes to solve larger instances of partition-coloring-type scheduling problems than pure MIP solvers can handle, providing a concrete route for scaling EV fleet optimization and related applications.

major comments (2)

[Computational results] Computational results section: the claim that QAIA variants 'prove optimality' on large instances where the Gurobi baseline terminates with positive gaps assumes that BSB and SimCIM always return exact maximum independent sets in every pricing call. Because these are iterative heuristics without certificates, the manuscript must supply explicit verification (e.g., comparison against an exact MIP solver on the encountered subproblems or a proof that the heuristics are exact on the tested graph family) for the optimality statements to be credible.
[Computational experiments] Instance generation and experimental setup: no description is given of how the synthetic EV instances and their conflict graphs are constructed, how QAIA hyperparameters are chosen, or what statistical tests (if any) support the performance differences. These omissions make it impossible to judge whether the observed outperformance is robust or an artifact of the particular test set.

minor comments (2)

[Abstract] The abstract and introduction should clarify the precise sizes (number of vehicles, intervals, density of conflict graphs) of the 'small', 'medium', and 'large' instance classes.
[Pricing subproblem] Notation for the QUBO reformulation of the maximum independent set pricing problem should be introduced with an explicit equation reference rather than left implicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and will revise the manuscript to improve clarity and rigor.

read point-by-point responses

Referee: [Computational results] Computational results section: the claim that QAIA variants 'prove optimality' on large instances where the Gurobi baseline terminates with positive gaps assumes that BSB and SimCIM always return exact maximum independent sets in every pricing call. Because these are iterative heuristics without certificates, the manuscript must supply explicit verification (e.g., comparison against an exact MIP solver on the encountered subproblems or a proof that the heuristics are exact on the tested graph family) for the optimality statements to be credible.

Authors: We agree that BSB and SimCIM are heuristics without built-in optimality certificates, so zero-gap termination of the branch-and-price procedure does not constitute a rigorous proof of optimality for the original problem. We will revise the computational results section to explicitly qualify all optimality claims, stating that the reported solutions and gaps are with respect to the columns generated by the QAIA solvers. In addition, we will add a verification subsection that solves a representative sample of the encountered pricing subproblems exactly with Gurobi and reports the frequency with which the QAIA heuristics recovered the optimal independent sets. This will allow readers to assess the practical reliability of the optimality statements. revision: yes
Referee: [Computational experiments] Instance generation and experimental setup: no description is given of how the synthetic EV instances and their conflict graphs are constructed, how QAIA hyperparameters are chosen, or what statistical tests (if any) support the performance differences. These omissions make it impossible to judge whether the observed outperformance is robust or an artifact of the particular test set.

Authors: We apologize for these omissions. We will insert a dedicated subsection on experimental setup that fully describes the synthetic instance generator (vehicle counts, time discretization, charger capacities, dwell-time distributions, and the precise rules used to build the conflict graph). We will also document the exact hyperparameter values employed for BSB and SimCIM, together with the tuning protocol. Finally, we will re-run the experiments with multiple random seeds, report means and standard deviations, and include paired statistical tests (Wilcoxon signed-rank) to quantify the significance of the observed performance differences. revision: yes

Circularity Check

0 steps flagged

No circularity: standard branch-and-price with external solver choice

full rationale

The paper describes a conventional branch-and-price decomposition for the partition coloring formulation of EV scheduling. The master problem is solved by Gurobi and the pricing subproblem (maximum independent set) is converted to QUBO and handed to QAIA heuristics (BSB, SimCIM) from the MindQuantum library. No equation or claim reduces to a fitted parameter, self-definition, or self-citation chain; the reported performance differences are empirical outcomes on synthetic instances rather than algebraic identities. The framework therefore contains no load-bearing step that is equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the modeling assumption that the conflict graph captures all relevant constraints and on the empirical behavior of the chosen QAIA solvers on the generated test set.

axioms (1)

domain assumption The conflict graph constructed from temporal and resource constraints accurately encodes feasible charging schedules.
Invoked when the EV problem is mapped to the partition coloring formulation.

pith-pipeline@v0.9.0 · 5565 in / 1240 out tokens · 48637 ms · 2026-05-15T01:01:15.037659+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

pricing subproblem is a maximum independent set problem. The latter is reformulated as a quadratic unconstrained binary optimization (QUBO) model and solved by quantum-annealing-inspired algorithms (QAIA) ... bSB and SimCIM
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We model this problem as a variant of the Partition Coloring Problem (PCP)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Scheduling electric vehicle regular charging tasks: A review of deterministic models.European Journal of Operational Research, 325(2):221–232, 2025

Alexandre Dolgui, Sergey Kovalev, and Mikhail Y Kovalyov. Scheduling electric vehicle regular charging tasks: A review of deterministic models.European Journal of Operational Research, 325(2):221–232, 2025

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Coordinating day-ahead and intraday scheduling for bidirectional charging of fleet evs.Automation, 6(4):64, 2025

Shiwei Shen, Syed Irtaza Haider, Razan Habeeb, and Frank HP Fitzek. Coordinating day-ahead and intraday scheduling for bidirectional charging of fleet evs.Automation, 6(4):64, 2025

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Optimizing trading of electric vehicle charging flexibility in the continuous intraday market under user and market uncertainties.Applied Energy, 381:125103, 2025

Raviteja Chemudupaty, Timothée Hornek, Ivan Pavi ´c, and Sergio Potenciano Menci. Optimizing trading of electric vehicle charging flexibility in the continuous intraday market under user and market uncertainties.Applied Energy, 381:125103, 2025

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Ji Wu, Hao Su, Jinhao Meng, and Mingqiang Lin. Electric vehicle charging scheduling considering infrastructure constraints.Energy, 278:127806, 2023

work page 2023

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Electric vehicle fleets: Scalable route and recharge scheduling through column generation.Transportation Science, 57(3):631–646, 2023

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